Prove that $a = b^m$ if $\frac{a^n-1}{b^n-1} \in \mathbb{Z}$ for $n \in \mathbb{N}.$

Asked Sep 25 '16 at 19:46

Active Sep 25 '16 at 21:44

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Few weeks ago I asked this problem in math.stackexchange.com. I got one answer, but soution mentioned in this answer uses only for big natural numbers. So thought about write this problem there. $$a = b^m$$ if $\displaystyle\frac{a^n-1}{b^n-1} \in \mathbb{Z} $ , $\forall n \in \mathbb{N}.$

This problem drives me crazy for a month. Help.

asked Sep 25 '16 at 19:46

openspace

6,470

1

$a=1+k(b^n-1)$? – Sep 25 '16 at 19:52
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Quantifiers are missing in your problem. – Sep 25 '16 at 19:53
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Various solutions and references are available at http://artofproblemsolving.com/community/c6t116f6h4556_a_masterpiece – Max Alekseyev Sep 25 '16 at 20:47
What does m equal to? – John Mayne Sep 25 '16 at 21:49
The earlier question was presumably http://math.stackexchange.com/questions/1919550/if-fracan-1bn-1-is-a-natural-number-for-every-n-then-a-bm, which was closed as a duplicate. – Gerry Myerson Sep 25 '16 at 22:59

Prove that $a = b^m$ if $\frac{a^n-1}{b^n-1} \in \mathbb{Z}$ for $n \in \mathbb{N}.$

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